\subsection*{2012-07-09}
We want to set up the hydro - gravity correspondence. We work in coordinates in which $z = 0$ is the AdS boundary. 
Expand the metric around $z$:
\begin{equation}
  g|_\sim \approx g_\partial + z^2 T_\partial + O(z^4)
\end{equation}  

The first $g_\partial$ is the non-normalizable thing, which is to say that we have to fix it -- choose it. The order $z^2$ is normalizable.

We want to choose flat background metric -- studying fluid in flat. 

We want a solution to Einstein equations such that the background boundary (first term) is flat, and the second equation is the "normalized" fluid stress-energy tensor:
\begin{equation}
  T_\partial = const(\rho + 3 u u) + \dots
\end{equation}

Given a metric, how do we know what is the corresponding fluid velocity? Go to to the boundary, compute the stress tensor $T_{\mu\nu}$, and find a boost velocity that takes you into the fluid frame! That is $T_\mu\nu v^\mu = 0$. The boost velocity is the fluid velocity, duh.

Turn this construction the other way around. 
Start with a 'nice' black hole solution to Einstein field equations. Symmetric, static, the usual good stuff. 
Here it is in (fill) coordinates:
\begin{equation}
  ds^2 = 2 dv dr - r^2 f(br) dv^2 + r^2 d\vec x^2
\end{equation}
with $f(r) = 1 - \tfrac{1}{r^3}$ and $b = \tfrac{3}{4\pi T}$.
Compute the boundary tensor, and realize this is a pretty boring solution -- the fluid is completely stationary. 

So let's play a trick -- boost. 
If fluid is stationary, move past it to make its velocity explicit in the form of the metric. 
Resulting metric:
\begin{equation}
  ds^2 = -2 u_\mu dx^\mu dr - r^2 f(br) u_\mu u_\nu dx^\mu dx^\nu + r^2 P_{\mu\nu} dx^\mu dx^\nu
\end{equation}
$u$ is a four vector of length 1, so this is a 3 parameter solution. 
Now comes the important step: having identified $u$ (and T) with the velocity of the fluid at the boundary, make it depend on the boundary coordinates. 
The resulting metric is NOT, generically, a solution to the einstein equations, and that's why we need to go through the painful expansion procedure to make sure it is.

The expansion proceeds in the gradients of the velocity and temperature. 
Solving Einstein eqns order by order constrains the form of $u$ and $T$, in turn adding higher order corrections to the boundary stress energy tensor. 

The long term goal. 
Nobody has an idea how to characterize turnbulent flows. 
What's characteristically true (generally, qualitatively) of the horizon when the fluid dual to it is turbulent? 
How do you answer that question? 
We could try to look at dfq's, but proving things about them is hard. 
But what we can do is LOOK at them. 
Solve the diff equations, build the gravity dual, once you have solutions, just look at lots of them soultions and start learning. 
Maybe you can learn what variables to measure. 
Maybe some special energy density. 
At horizon? 
Maybe mean extrinsic curvature at the horizon? What about the Wilson lines of the fluid velocity. 

\subsubsection*{Questions for next time}
\begin{itemize}
\item Hydrodynamic Expansion (derivation)
\item Turbulent flow -- when is hydro justified, when is it not. 
\item What are the transport coefficient, physically. 
\item Role of the Navier Stokes Eq.  Viscous effects in regulating the Navier Stokes equation. 
\item Example: simple 1-component fluid in flat 3+1 Spacetime. EOMs @ 1st \& 2nd order hydro. 
\end{itemize}


\subsubsection*{References}
\begin{itemize}
\item Raamsdonk -- Black hole dynamics from atmospheric science. The '2+1' paper. Brief intro + technical derivation.
\item Bhattacharya, Hubeny, Minwalla, Rangamani -- Nonlinear Fluid Dynamics from Gravity. The '3+1' paper. Includes very general discussion about how the expansion of EFE is set up.
\item Rangamani -- Gravity \& Hydrodynamics Lectures. Good place to start with fluid/grav imho.
\item Hubeny, Minwalla, Rangamani -- The fluid gravity correspondence. Extremely well-written, a lot of insights. 
\item Andersson -- Relativistic fluid dynamics: physics for many different scales. Intro to hydro, good place to start in general
\item Gourgoulhon -- An introduction to rel hydro. Coordinate free intro to rel hydro and relativity in general.
\end{itemize}
